Evaluate the Integral

Problem

(∫_7^8)(x√(,x−7)*d(x))

Solution

1. Identify the substitution to simplify the integrand. Let u=x−7

2. Differentiate the substitution to find d(u) Since u=x−7 then d(u)=d(x)

3. Express x in terms of u From u=x−7 we get x=u+7

4. Change the limits of integration. When x=7 u=7−7=0 When x=8 u=8−7=1

5. Substitute these into the integral to rewrite it in terms of u

(∫_0^1)((u+7)√(,u)*d(u))

6. Distribute the √(,u) (which is u(1/2) into the parentheses.

(∫_0^1)((u(3/2)+7*u(1/2))*d(u))

7. Integrate each term using the power rule (∫_^)(un*d(u))=(u(n+1))/(n+1)

[2/5*u(5/2)+7⋅2/3*u(3/2)]10

8. Simplify the coefficients.

[2/5*u(5/2)+14/3*u(3/2)]10

9. Evaluate the definite integral at the upper and lower limits.

(2/5*(1)(5/2)+14/3*(1)(3/2))−(2/5*(0)(5/2)+14/3*(0)(3/2))

10. Calculate the final numerical value by finding a common denominator.

2/5+14/3=6/15+70/15=76/15

Final Answer

(∫_7^8)(x√(,x−7)*d(x))=76/15

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